Learn Hardcore German: Der Trainer erinnert uns daran, rechtzeitig ins Fitnessstudio zu kommen, damit die Stunde pünktlich beginnen kann. - The coach reminds us to come to the gym on time so that the class can start punctually.

In our sentence the “something” is a whole action: rechtzeitig ins Fitnessstudio zu kommen.
German normally uses a pronominal adverb (daran) before a zu + infinitive clause:

jemanden daran erinnern, etwas zu tun
→ to remind someone to do something

So:

Der Trainer erinnert uns daran, rechtzeitig ins Fitnessstudio zu kommen
= The trainer reminds us to get to the gym on time.

Without daran, the sentence sounds incomplete or at least less natural in standard German.

What is the difference between jemanden erinnern and sich erinnern?

Two different structures:

jemanden an etwas erinnern
- non‑reflexive
- meaning: to remind someone of something
- example: Der Trainer erinnert uns daran.
sich an etwas erinnern
- reflexive
- meaning: to remember something
- example: Wir erinnern uns daran. = We remember that.

So:

Der Trainer erinnert uns daran = The trainer reminds us of it.
Wir erinnern uns daran = We remember it.

Why is it ins Fitnessstudio and not im Fitnessstudio or in das Fitnessstudio?

ins = in + das (contraction)
ins Fitnessstudio uses the accusative because there is movement towards a place (into the gym).

German uses:

Akkusativ (Wohin? – where to?) for direction:
- ins Fitnessstudio gehen – to go into the gym
Dativ (Wo? – where?) for location:
- im Fitnessstudio sein – to be in the gym

in das Fitnessstudio is grammatically OK but in everyday speech you almost always say ins Fitnessstudio.
im Fitnessstudio (in dem Fitnessstudio) would mean already in the gym, not going to the gym.

What is the function of rechtzeitig here, and how is it different from pünktlich?

Both relate to time, but they are not the same:

rechtzeitig ≈ in good time / early enough (before something starts)
- rechtzeitig ins Fitnessstudio kommen
  = arrive early enough, not at the last second.
pünktlich ≈ on time / at the exact, agreed time
- die Stunde beginnt pünktlich
  = the class starts exactly at the scheduled time.

So the trainer reminds us to come early enough (rechtzeitig), so that the class can start exactly on time (pünktlich).

What does damit do in this sentence, and how does it relate to English?

damit introduces a purpose clause = so that / in order that.

Structure:

Main clause: Der Trainer erinnert uns daran, rechtzeitig ins Fitnessstudio zu kommen,
Purpose clause: damit die Stunde pünktlich beginnen kann.

Meaning:
The trainer reminds us to come on time for the purpose that the class can start punctually.

In English, damit is usually translated as so that or in order that.

Could we replace damit with um … zu here?

No, not in this exact sentence, because um … zu requires the same subject in both clauses.

um … zu:
- same subject does the main clause and the zu-infinitive action.
- Example:
  - Ich gehe früher, um pünktlich anzukommen.
    (I = the one who goes earlier and arrives punctually)
damit:
- can be used when the subject changes.
- In our sentence:
  - Main clause subject: Der Trainer
  - Purpose clause subject: die Stunde

So you cannot say:
✗ … rechtzeitig ins Fitnessstudio zu kommen, um die Stunde pünktlich beginnen zu können.
That would wrongly make uns (or implicitly we) the subject of beginnen (to begin the class).

Correct is:

…, damit die Stunde pünktlich beginnen kann.
because die Stunde is the subject of beginnen.

Why is the word order beginnen kann at the end, and not kann beginnen?

In a subordinate clause (introduced by damit), all verbs go to the end of the clause.

Rule for verb order with a modal verb in a subordinate clause:

Main (lexical) verb in infinitive
Modal verb (also in infinitive form)

So we get:

damit die Stunde pünktlich beginnen kann

In a main clause, the order is different:

Die Stunde kann pünktlich beginnen.

Compare:

Main clause: Die Stunde kann pünktlich beginnen.
Subordinate: …, damit die Stunde pünktlich beginnen kann.

What is the role of zu kommen here? Is it a special tense?

zu kommen is not a special tense. It’s a zu-infinitive construction:

rechtzeitig ins Fitnessstudio zu kommen
acts like a noun phrase / object, something like “coming to the gym on time.”

Structure:

jemanden daran erinnern, etwas zu tun
- jemanden = object (here uns)
- daran = refers forward to the action
- etwas zu tun = zu + infinitive clause (here rechtzeitig ins Fitnessstudio zu kommen)

So grammatically, the trainer reminds us of this action: coming to the gym on time.

Why is it Der Trainer and uns here? What cases are being used?

Der Trainer – nominative singular, masculine
- subject of the main clause (the one doing the reminding)
uns – accusative (1st person plural pronoun)
- direct object (the ones being reminded)

The verb jemanden erinnern always takes a direct object in the accusative:

Der Trainer (NOM) erinnert uns (AKK).

Can we leave out daran and just say:
Der Trainer erinnert uns, rechtzeitig ins Fitnessstudio zu kommen?

Native speakers do sometimes drop daran in casual speech, and many will understand it, but:

The standard, idiomatic pattern is
jemanden daran erinnern, etwas zu tun.

Without daran, the sentence sounds:

less clear (What exactly is connected to erinnern?)
less natural in careful written German.

So for learners and in correct written German you should keep:

Der Trainer erinnert uns daran, rechtzeitig ins Fitnessstudio zu kommen.

Why is there a comma before damit?

Because damit starts a subordinate clause.

German punctuation rule:

A subordinate clause (Nebensatz) is always separated by a comma from the main clause.

So:

…, rechtzeitig ins Fitnessstudio zu kommen,
damit die Stunde pünktlich beginnen kann.

Both the zu-infinitive clause and the damit-clause are set off by commas.

Does die Stunde literally mean “the hour,” or does it mean “class” here?

Literally, die Stunde = the hour, but in many contexts it means:

class / lesson / session

In the context of a trainer and a gym, die Stunde is naturally understood as:

the (workout) class
or
the training session

So the meaning is:
… so that the class can start on time.

Why is it pünktlich beginnen kann and not just pünktlich beginnt?

Both are possible, but they have slightly different nuances:

damit die Stunde pünktlich beginnt
- more straightforward: so that the class starts on time.
damit die Stunde pünktlich beginnen kann
- emphasizes the possibility/ability:
  so that the class is able to start on time / so that it can start on time.

Using kann highlights that the start of the class depends on people arriving on time.